Hahn-Banach and Dual Spaces - Examples and Constructions

Geometric forms of the Hahn-Banach Theorem provide powerful tools for separating and supporting convex sets, with applications throughout optimization and economics.

TheoremHahn-Banach Separation Theorem

Let $X$ be a normed space and $A, B \subset X$ disjoint non-empty convex sets.

Weak Separation: If $A$ is open, there exist $\phi \in X^* \setminus \{0\}$ and $\alpha \in \mathbb{R}$ such that $\phi(a) \leq \alpha \leq \phi(b)$ for all $a \in A$ and $b \in B$
Strong Separation: If $A$ is closed, $B$ is compact, there exist $\phi \in X^*$ and $\alpha, \beta \in \mathbb{R}$ with $\alpha < \beta$ such that $\phi(a) < \alpha < \beta < \phi(b)$ for all $a \in A$ and $b \in B$

These separation theorems provide a geometric interpretation of the Hahn-Banach Theorem: convex sets can be separated by continuous linear functionals (hyperplanes).

DefinitionSupporting Functional

Let $C \subset X$ be a convex set and $x_0 \in \partial C$ a boundary point. A functional $\phi \in X^*$ is a supporting functional for $C$ at $x_0$ if $\phi(x_0) = \sup_{x \in C} \phi(x)$

The hyperplane $\{x : \phi(x) = \phi(x_0)\}$ is called a supporting hyperplane.

TheoremExistence of Supporting Functionals

Let $C$ be a non-empty closed convex set in a normed space $X$ and $x_0 \in \partial C$ . Then there exists a non-zero supporting functional $\phi \in X^*$ for $C$ at $x_0$ .

ExampleApplications

Optimization: The subdifferential of a convex function at a point consists of supporting functionals to its epigraph
Economics: The separation theorem underlies the fundamental theorems of welfare economics
Game Theory: Supporting hyperplanes are used to prove the minimax theorem
Variational Inequalities: Solutions are characterized by supporting functionals

ExampleMinkowski Functional

Let $C \subset X$ be a convex set containing $0$ in its interior. The Minkowski functional of $C$ is $p_C(x) = \inf\{t > 0 : x \in tC\}$

This is a sublinear functional. If $C$ is balanced (i.e., $\alpha C \subset C$ for all $|\alpha| \leq 1$ ), then $p_C$ is a seminorm. The Hahn-Banach Theorem can be proven using Minkowski functionals.

Remark

The geometric versions of Hahn-Banach are essential tools in convex analysis and optimization. They show that the analytic concept of linear functionals has deep geometric meaning in terms of separating and supporting convex sets.

These results form the foundation for duality theory in optimization, where primal and dual problems are related through separating hyperplanes.